Warm up Add the following polynomials x + 2y = 10 5y – x = 7 + 4x – 3y = 1 + 9y + 4x = -1

Objective The student will be able to: solve systems of equations using elimination with addition and subtraction.

Solving Systems of Equations So far, we have solved systems using graphing and substitution. These notes show how to solve the system algebraically using ELIMINATION with addition and subtraction. Elimination is easiest when the equations are in standard form.

Solving a system of equations by elimination using addition and subtraction. Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Add the equations. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Standard Form: Ax + By = C Look for variables that have the same coefficient. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations.

Example 1 x + y = 5 3x – y = 7 Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. They already are! The y’s have the same coefficient. Step 3: Add or subtract the equations. Add to eliminate y. x + y = 5 (+) 3x – y = 7 4x = 12 x = 3

Example 1 Continued Step 4: Plug back in to find the other variable. x + y = 5 (3) + y = 5 y = 2 Step 5: Check your solution. (3, 2) (3) + (2) = 5 3(3) - (2) = 7 The solution is (3, 2). What do you think the answer would be if you solved using substitution? x + y = 5 3x – y = 7

Example 2 4x + y = 7 4x – 2y = -2 Step 1: Put the equations in Standard Form. They already are! Step 2: Determine which variable to eliminate. The x’s have the same coefficient. Step 3: Add the equations. Note, since both 4x’s are positive, when we add them, they will not cancel out. So, we have to first do one more step: multiply one of the equations by -1.

Example 2 Continued 4x + y = 7 4x – 2y = -2 I will multiply the second equation by -1, so those negative signs will become positive. (-1) * 4x – 2y = -2 = -4x + 2y = 2 (every sign is switched) Now I will add this new equation to the first equation.

Example 2 Continued 4x + y = 7 4x – 2y = -2 Now we are ready for step 3, using the new equation Step 3: Add the equations. 4x + y = 7 (+) -4x + 2y = 2 3y = 9 y=3

Example 2 Continued. Step 4: Plug back in to find the other variable. 4x + y = 7 4x + (3) = 7 4x = 4 x = 1 Step 5: Check your solution. (1, 3) 4(1) + (3) = 7 4(1) - 2(3) = -2 4x + y = 7 4x – 2y = -2

Which step would eliminate a variable? 1.Isolate y in the first equation 2.Add the equations 3.Subtract the equations 4.Multiply the first equation by -4 3x + y = 4 3x + 4y = 6

You Try! Solve using elimination. 1.(2, 2) 2.(9, 3) 3.(4, 5) 4.(5, 4) 2x – 3y = -2 x + 3y = 17

Example 3 (Here we will have to put it into standard form) y = 7 – 2x 4x + y = 5 Step 1: Put the equations in Standard Form. 2x + y = 7 4x + y = 5 Step 2: Determine which variable to eliminate. The y’s have the same coefficient. Step 3: Add the equations. Subtract to eliminate y. 2x + y = 7 (-) 4x - y = -5 -2x = 2 x = -1

Example 3 continued Step 4: Plug back in to find the other variable. y = 7 – 2x y = 7 – 2(-1) y = 9 Step 5: Check your solution. (-1, 9) (9) = 7 – 2(-1) 4(-1) + (9) = 5 y = 7 – 2x 4x + y = 5

What is the first step when solving with elimination? 1.Add or subtract the equations. 2.Plug numbers into the equation. 3.Solve for a variable. 4.Check your answer. 5.Determine which variable to eliminate. 6.Put the equations in standard form.

Find two numbers whose sum is 18 and whose difference 22. Challenge Problem! The sum of two numbers is 70 and their difference is 24. Find the two numbers.

Solution The sum of two numbers is 70 and their difference is 24. Find the two numbers. x+y=70 (47)+y=70 + x-y=24 y= x=94 x=47 y=23 Solution (47,23)

Homework s/Alg1Worksheets/Systems%20of%20Equation s%20Elimination.pdf s/Alg1Worksheets/Systems%20of%20Equation s%20Elimination.pdf Problems 1-8